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DESY 06-230 Neutrino Dark Energy – Revisiting the Stability Issue Ole Eggers Bjælde1, Anthony W. Brookfield2, Carsten van de 8 0 Bruck3, Steen Hannestad1, David F. Mota4,5, Lily Schrempp6, and 0 2 Domenico Tocchini-Valentini7 n 1 Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, a DK-8000 Aarhus C, Denmark J 2 Department of Applied Mathematics and Department of Physics, Astro-Particle 7 1 Theory & Cosmology Group, Hounsfield Road, Hicks Building, University of Sheffield, Sheffield S3 7RH, UK ] h 3 Department of Applied Mathematics, Astro-Particle Theory & Cosmology Group, p Hounsfield Road, Hicks Building, University of Sheffield, Sheffield S3 7RH, UK o- 4 Institute for Theoretical Physics, University of Heidelberg, D-69120 Heidelberg, r Germany t s 5 Institute of Theoretical Astrophysics, University of Astrophysics, N-0315 Oslo, Norway a [ 6 Deutsches Elektron-SynchrotonDESY, Hamburg, Notkestr. 85, 22607 Hamburg, Germany 2 7 Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD v 8 21218,USA 1 0 E-mail: [email protected],[email protected], 2 [email protected],[email protected], 5. [email protected],[email protected], 0 [email protected] 7 0 : Abstract. A coupling between a light scalar field and neutrinos has been widely v i discussed as a mechanism for linking (time varying) neutrino masses and the present X energy density and equation of state of dark energy. However, it has been pointed out r that the viability of this scenario in the non-relativistic neutrino regime is threatened by a the strong growth of hydrodynamic perturbations associated with a negative adiabatic sound speed squared. In this paper we revisit the stability issue in the framework of linear perturbation theory in a model independent way. The criterion for the stability of a model is translated into a constraint on the scalar-neutrino coupling, which depends on the ratio of the energy densities in neutrinos and cold dark matter. We illustrate our results by providing meaningful examples both for stable and unstable models. PACS numbers: 13.15.+g,64.30.+t, 64.70.Fx, 98.80.Cq Neutrino Dark Energy – Revisiting the Stability Issue 2 1. Introduction Precision observations of the cosmic microwave background [1–3], the large scale structure of galaxies [4], and distant type Ia supernovae [5–8] have led to a new standard model of cosmology in which the energy density is dominated by dark energy with negative pressure, leading to an accelerated expansion of the universe. The simplest possible explanation for dark energy is the cosmological constant which has P = wρ with w = 1 at all times. However, since the cosmological constant has − a magnitude completely different from theoretical expectations one is naturally led to consider other explanations for the dark energy. A light scalar field rolling in a very flat potential would for instance be a candidate better motivated from high energy physics [9–11]. In the limit of a completely flat potential it would have w = 1. Such models are − generically known as quintessence models [12–17]. The scalar field is usually assumed to be minimally coupled to matter and to curvature, but very interesting effects can occur if this assumption is relaxed (see for instance [18–24]). In general such models alleviate the required fine tuning in order to achieve Ω Ω , where Ω and Ω are the dark energy X m X m ∼ and matter densities at present. Also by properly choosing the quintessence potential it is possible to achieve tracking behaviour of the scalar field sothat onealso avoids the extreme fine tuning of the initial conditions for the field. Many other possibilities have been considered, like k-essence, which is essentially a scalar field with a non-standard kinetic term [25–27]. It is also possible, although not without problems, to construct models which have w < 1, the so-called phantom dark − energy models [28–30]. Finally, there are even more exotic models where the cosmological acceleration is not provided by dark energy, but rather by a modification of the Friedmann equation due to modifications of gravity on large scales [31,32], or even due to higher order curvature terms in the gravity Lagrangian [33–35]. Avery interesting proposalistheso-calledmassvarying neutrino (MaVaN)model[36– 38] in which a light scalar field couples to neutrinos. Due to the coupling, the mass of the scalar field does not have to be as small as the Hubble scale but can be much larger, while the model still accomplishes late-time acceleration. This scenario also holds the interesting possibility of circumventing the well-known cosmological bound on the neutrino mass [3,4,40–51]. The scenario is a variant of the chameleon cosmology model [52–54] in which a light scalar field couples democratically to all non-relativistic matter. The idea in the MaVaN model is to write down an effective potential for the scalar field which as a result of the coupling contains a term related to the neutrino energy density. If the pure scalar field potential is tuned appropriately the effective potential including the neutrino contribution will have a minimum with a steep second derivative for some finite scalar field VEV. The scalar field is therefore locked in the minimum and when the minimum evolves due to changing neutrino energy density the field tracks this evolution adiabatically. This naturally leads to a dynamical effective equation of state for the combined scalar - neutrino fluid close to w = 1 today, and to a neutrino mass − which is related to the combined neutrino-scalar field fluid’s energy density ρ . Since DE ρ decreases with time, also the neutrino mass varies in this kind of scenario, where its DE Neutrino Dark Energy – Revisiting the Stability Issue 3 1/4 present value is explained in terms of ρ (a = 1). Possible tests for the MaVaN scenario DE can be found in Ref. [55–62]. MaVaN models, however, suffer from the problem that for some choices of scalar- neutrino couplings and scalar field potentials the combined fluid is subject to an instability once the neutrinos become non–relativistic. Effectively the scalar field mediates an attractive force between neutrinos which can possibly lead to the formation of neutrino nuggets [63]. This in turn would make the combined fluid behave like cold dark matter and thus render it non-viable as a candidate for dark energy. In perturbation theory the formation of these nuggets can be seen as a consequence of an imaginary speed of sound for the combined fluid, signaling fast growth of instabilities. However, an imaginary speed of sound cannot be generally used as a sufficient criterion for the instabilities, as the drag provided by cold dark matter may postpone those instabilities. Theinstabilitycanpossiblyoccurinthesemodelsbecausetheeffectivemassassociated with the scalar field is much larger than H. Accordingly, on sub-Horizon scales larger than the effective Compton wavelength of the scalar field m−1 < a/k < H−1 the perturbations φ are adiabatic. This is a consequence of the steepness of the effective potential and can be remedied by making the potential sufficiently flat. In this case the evolution of the field is highly non-adiabatic [64,65]. However, this model has the disadvantage that the neutrino mass is no longer related naturally to the dark energy density and equation of state. In this paper we study various choices of scalar-neutrino couplings and scalar field potentials with the aim of identifying the conditions for the instability to occur. In the next section wereview theformalismneeded tostudy massvarying neutrinosandinsection 3 we derive the equation of motion of the neutrino perturbations. Section 4 contains our results for various couplings and potentials, and finally section 5 contains a discussion and conclusion. 2. Formalism The idea in the so-called Mass Varying Neutrino (MaVaN) scenario [36–38] is to introduce a coupling between (relic) neutrinos and a light scalar field and to identify this coupled fluid with dark energy. As a direct consequence of this new interaction, the neutrino mass m is generated from the vacuum expectation value (VEV) of the scalar field and becomes ν linked to its dynamics. Thus the pressure P (m (φ),a) and energy density ρ (m (φ),a) of ν ν ν ν the uniform neutrino background contribute to the effective potential V(φ,a) of the scalar field. The effective potential is defined by V(φ) = V (φ)+(ρ 3P ) (1) φ ν ν − where V (φ)denotes thefundamentalscalar potentialandaisthescalefactor. Throughout φ the paper we assume a flat Friedman-Robertson-Walker cosmology and use the convention a = 1, where we take the subscript 0 to denote present day values. 0 Assuming the neutrino distribution to be Fermi-Dirac and neglecting the chemical potential, theenergydensity andpressure oftheneutrinoscanbeexpressed inthefollowing Neutrino Dark Energy – Revisiting the Stability Issue 4 form [39] dyy2 y2+ m2ν(φ) ρ (a,φ) = Tν4(a) ∞ r Tν2(a) , ν π2 ey +1 Z0 T4(a) ∞ dyy4 P (a,φ) = ν , (2) ν 3π2 Z0 y2+ m2ν(φ)(ey +1) Tν2(a) r whereT = T /aistheneutrinotemperatureandy correspondstotheratiooftheneutrino ν ν0 momentum and neutrino temperature, y = p /T . ν ν The energy density and pressure of the scalar field are given by the usual expressions, 1 ρ (a) = φ˙2 +V (φ), φ 2a2 φ 1 P (a) = φ˙2 V (φ). (3) φ 2a2 − φ Defining w = P /ρ to be the equation of state of the coupled dark energy fluid, DE DE where P = P +P denotes its pressure and ρ = ρ +ρ its energy density, and the DE ν φ DE ν φ requirement of energy conservation gives, ρ˙ +3Hρ (1+w) = 0. (4) DE DE Here H a˙ and we use dots to refer to the derivative with respect to conformal time. ≡ a Taking Eq. (4) into account, one arrives at a modified Klein-Gordon equation describing the evolution of φ, φ¨+2Hφ˙ +a2V′ = a2β(ρ 3p ). (5) φ − ν − ν ′ Here and in the following primes denote derivatives with respect to φ ( = ∂/∂φ) and β = dlogmν is the coupling between the scalar field and the neutrinos. dφ 2.1. The fully adiabatic case In the following let us consider the late time evolution of the coupled scalar-neutrino fluid in the limit m T where the neutrinos are non-relativistic. It is in this regime that ν ν ≫ MaVaN models can potentially become unstable for the following reason: The attractive force mediated by the scalar field (which can be much stronger than gravity) acts as a driving force for the instabilities. But as long as the neutrinos are still relativistic, the evolution of the density perturbations will be dominated by pressure which inhibits their growth, as the strength of the coupling is suppressed when ρ = 3P . ν ν In the non-relativistic limit m T , the expressions for the energy density and ν ν ≫ pressure in neutrinos in Eq. (2) reduce to ρ m n , ν ν ν ≃ P 0, (6) ν ≃ such that Eq. (1) takes the form V = ρ +V = m n +V . (7) ν φ ν ν φ Neutrino Dark Energy – Revisiting the Stability Issue 5 Assuming the curvature scale of the potential and thus the mass of the scalar field m φ to be much larger than the expansion rate of the Universe, V′′ = ρ β′ +β2 +V′′ m2 H2, (8) ν φ ≡ φ ≫ (cid:16) (cid:17) the adiabatic solution to the equation of motion of the scalar field in Eq. (5) applies [38] . ‡ As a consequence, the scalar field instantaneously tracks the minimum of its effective potential V, solution to the condition ∂ρ ∂V ∂V ′ ′ ′ ′ ν φ ′ φ V = ρ +V = m + = m n + = 0. (9) ν φ ν ∂mν ∂mν! ν ν ∂mν! As the universe expands the neutrino energy density gets diluted, thus naturally giving rise to a slow evolution of V(φ). Consequently, the value of the scalar field φ evolves on ′ cosmological time scales. Note that as long as m does not vanish, this implies that also ν the neutrino mass m (φ) is promoted to a time dependent, dynamical quantity. Its late ν time evolution can be determined from the last equality in Eq. (9). In order to specify good candidate potentials V (φ) for a viable MaVaN model of dark φ energy, we must demand that the equation of state parameter w of the coupled scalar- neutrino fluid today roughly satisfies w 1 as suggested by observations [66]. By noting ∼ − that for constant w at late times, ρ V a−3(1+w) (10) DE ∼ ∝ and by requiring energy conservation according to Eq. (4), one arrives at [38] 1∂log V 1+w = . (11) −3 ∂loga In the non-relativistic limit m T this is equivalent to ν ν ≫ ′ ′ a ∂n ∂m V m V 1+w = m ν +n ν + φ = ν φ, (12) −3V ν ∂a ν ∂a a′ ! −m′V ν ′ where in the last equality it has been used that V = 0 according to Eq. (9). To allow for an equation of state close to w 1 today one can conclude that either the scalar ∼ − potential V has to be fairly flat or the dependence of the neutrino mass on the scalar field φ has to be very steep. 2.2. The general case As it will turn out later, the influence of the cosmic expansion in combination with the gravitational drag exerted by CDM on the neutrinos can have an effect on the stability of a MaVaN model. However, to begin we will neglect any growth-slowing effects on the perturbations and proceed with a more general analysis of this case. Under these circumstances, the dynamics of the perturbations are solely determined by the sound speed squared which for a general fluid component i takes the following form, δP c2 = i, (13) si δρ i Inthiscasefor φ <M 3 1018GeVtheeffectsofthekineticenergytermscanbesafelyignored[38]. pl ‡ | | ≃ × Neutrino Dark Energy – Revisiting the Stability Issue 6 where P and ρ denote the fluid’s pressure and energy density, respectively. The sound i i speed c2 can be expressed in terms of the sound speed c2 arising from purely adiabatic si ai perturbations as well as an additional entropy perturbation Γ and the density contrast i δ = δρ /ρ in the given frame [67,68], i i i w Γ = (c2 c2 )δ , (14) i i si − ai i ˙ P δP δρ i i i = . (15) ρi P˙i − ρ˙i ! Here w denotes the equation of state parameter and Γ is a measure for the relative i i displacement between hypersurfaces of uniform pressure and uniform energy density. For most dark energy candidates (like quintessence or k-essence) dissipative processes evoke entropy perturbations and thus Γ = 0. i 6 However, inMaVaNmodelstheeffectivemassofthescalarfieldm H setsthescale, φ ≫ m−1, where these processes andtheassociated gradient terms become unimportant [63,69], φ to be much smaller than the Hubble radius (in contrast to a quintessence field with finely-tuned mass <H and long range >H−1). As a consequence, on sub-Hubble scales H−1 > a > m−1 all∼dynamical properties∼of (non-relativistic) MaVaNs are set by the local k φ neutrino energy density [63]. In particular, for small deviations away from the minimum of its effective potential, the scalar field re-adjusts to its new minimum on time scales m−1 small compared to the characteristic cosmological time scale H−1. In this case the ∼ φ hydrodynamic perturbations inMaVaNsareadiabatic. This means thesystem of neutrinos and the scalar field can be treated as a unified fluid with pressure P = P + P and DE ν φ energy density ρ = ρ +ρ without intrinsic entropy, Γ = 0 . DE ν φ DE § If any growth-slowing effects can be neglected, the perturbations in a MaVaN model are driven by the effective sound speed squared given by P˙ w˙ρ +wρ˙ w˙ c2 = DE = DE DE = w , (16) a ρ˙ ρ˙ − 3H(1+w) DE DE where Eq. (4) and Eq. (15) have been used. In the case c2 > 0 the attractive scalar force is a offset bypressure forcesandthefluctuationsoscillate assoundwaves andcanbeconsidered as stable. However, for c2 < 0 perturbations become unstable and tend to blow up. a 3. Evolution of the Perturbations In this section we will analyse the linear MaVaN perturbations in the synchronous gauge, which is characterised by a perturbed line element of the form ds2 = a(τ)2( dτ2 +(δ +h )dxidxj), (17) ij ij − where τ denotes conformal time and h is the metric perturbation. Here and in the ij following dots represent derivatives with respect to τ. Most of our other notations and (see also [70] for another example of unified models) § Neutrino Dark Energy – Revisiting the Stability Issue 7 conventions comply with those in Ma and Bertschinger [74]. Consequently, the Friedmann equation takes the form a2 φ˙2 3H2 = +V (φ)+ρ , (18) M2 2a2 φ m! pl with M (√8πG)−1 denoting the reduced Planck mass and the subscript m comprising pl ≡ all matter species. Since the following perturbation equations have been widely discussed in the literature (e.g. [23,53,65,76,77] and references therein), we will simply state them here for neutrinos coupled to a scalar field. The evolution equation for the MaVaN density contrast δ = δρ /ρ is given by [65], ν ν ν ˙ δp h ˙ ˙ ν δ = 3 H +βφ w δ (1+w ) θ + ν ν ν ν ν − δρν! − 2! (cid:16) (cid:17) ˙ ′ ˙ + β(1 3w )δφ+β φδφ(1 3w ), (19) ν ν − − where β = dlogmν. dφ Furthermore, the trace of the metric perturbation, h δijh , according to the ij ≡ linearised Einstein equations satisfies, a2 h¨ +Hh˙ = [δT0 δTi], where (20) M2 0 − i pl 1 δT0 = φ˙δφ˙ V′(φ)δφ ρ δ , (21) 0 − a2 − φ − m m m X 3 δTi = φ˙δφ˙ 3V′(φ)δφ+ ρ δ +3c2ρ δ +3c2ρ δ . (22) i a2 − φ r r b b b ν ν ν r X Here δTµ denotes the perturbed stress energy tensor and the subscripts m and r collect ν neutrinos, radiation, CDM and baryons (with sound speed c ) as well as (relativistic) b neutrinos and radiation, respectively. The evolution equation for the neutrino velocity perturbation θ ik vi with ν ≡ i ν vi dxi/dτ reads [65], ν ≡ δpν w˙ θ˙ = H(1 3w )θ ν θ + δρν k2δ ν ν ν ν ν − − − 1+w 1+w ν ν 1 3w + β − νk2δφ β(1 3w )φ˙θ k2σ , (23) ν ν ν 1+w − − − ν where σ denotes the neutrino shear as defined in [74]. ν Finally, the perturbed Klein-Gordon equation for the coupled scalar field is given by [65] 1 δ¨φ+2Hδ˙φ+ k2 +a2 V′′ +β′(ρ 3P ) δφ+ h˙φ˙ = (24) φ ν − ν 2 h n oi δp a2βδ ρ (1 3 ν). ν ν − − δρ ν We note that instead of proceeding via the fluid equations, Eqs. (19) and (23), the evolution of the neutrino density contrast can be calculated from the Boltzmann equation Neutrino Dark Energy – Revisiting the Stability Issue 8 [74]. We have verified analytically and numerically that the two methods yield identical results provided that the scalar-neutrino coupling is appropriately taken account of in the Boltzmann hierarchy [75]. As discussed in sec. 2 MaVaNs models can only possibly become unstable on sub- Hubble scales m−1 < a/k < H−1 in the non-relativistic regime of the neutrinos, where the φ perturbations evolve adiabatically. For our numerical results in the next section we solve the coupled Eqs. (19-24) in the (quasi-)adiabatic regime by neglecting the neutrino shear σ . This approximation is justified, since the scalar-neutrino coupling becomes important ν inthisregimeandm ismuchlargerthanthemeanmomentumoftheneutrinodistribution. ν For the purpose of gaining further analytical insight into the evolution of the neutrino density contrast, it is instructive to apply additional approximations to Eqs. (19-24) to be justified in the following. Since the minimum of the effective potential tracked by the scalar field evolves only slowly due to changes in the neutrino energy density, we can safely ignore terms ˙ proportional to φ. Moreover, in the non-relativistic regime of the neutrinos on scales m−1 < a/k < H−1, as a consequence of P 0 it follows that σ 0 and w 0 as φ ν ∼ ν ∼ ν ∼ well as ρ c2 0. In addition, in the following we substitute δφ by its average value r ∼ b ∼ corresponding to the forcing term on the right hand side of Eq. (24) in the above limits, βρ δ ¯ ν ν δφ = , (25) −(V′′ +ρ β′)+ k2 φ ν a2 which solves the perturbed Klein-Gordon equation reasonably well on all scales [23,76]. Finally, by combining the derivative of Eq. (19) with Eq. (20) – Eq. (23) and Eq. (25) in the non-relativistic limit, we arrive at the equation of motion for the neutrino density contrast valid at late times on length scales m−1 < a/k < H−1, φ δp 3 G 3 δ¨ +Hδ˙ + νk2 H2Ω eff δ = H2 Ω δ +Ω δ ν ν ν ν CDM CDM b b δρν − 2 G ! 2 " # (26) where 2β2M2 pl G = G 1+ and (27) eff  1+ a2 V′′ +ρ β′  k2{ φ ν } a2ρ  i Ω = . (28) i 3H2M2 pl Since neutrinos not only interact through gravity, but also through the force mediated by the scalar field, they feel an effective Newton’s constant G as defined in Eq. (27). eff The force depends upon the MaVaN model specific functions β and V and takes values φ between G and G(1 + 2β2M2) on very large and small length scales, respectively. The pl ′′ ′ −1 scale dependence of Geff is due to the finite range of the scalar field (Vφ +ρνβ ) 2, which according to Eq. (8) is equal to (m2φ − β2ρν)−21. For moderate coupling strength it is essentially given by the inverse scalar field mass, whereas for β 1/M it can take larger pl ≫ values. Accordingly, in a MaVaN model both the scalar potential V and the coupling φ Neutrino Dark Energy – Revisiting the Stability Issue 9 β influence the range of the scalar field force felt by neutrinos, whereas its strength is determined by the coupling β. The evolution of perturbations in cold dark matter (CDM) coupled to a light scalar field in coupled quintessence [23] and chameleon cosmologies [53] is governed by an equation similar to Eq. (26). However, we would like to point out that for the same coupling functions the dynamics of the perturbations in neutrinos can be quite different from those in coupled CDM. This is a result of the fact that Ω (Ω +Ω ). Whereas ν CDM b ≪ Ω 0.2 and Ω 0.05 [4] at present, Ω depends on the so far not known absolute CDM b ν ∼ ∼ neutrino mass scale realised in nature. Taking as a lower bound the mass splitting deduced from atmospheric neutrino flavour oscillation experiments and the upper bound derived from the Mainz tritium beta-decay experiments [81], we get 10−4<Ω <0.15 today . It ν k ∼ ∼ is important to note that since in the standard MaVaN scenario the neutrino mass is an increasing functionoftime, atearlier timestheratioΩ /(Ω +Ω )waseven smaller than ν CDM b today. In general it follows that the smaller this ratio is, the larger the relative influence of the forcing term on the RHS of Eq. (26) becomes. The forcing term describes the effect of the perturbations of other cosmic components on the dynamics of the neutrino density contrast and competes with the scalar field dependent term GeffΩ δ on the ∝ G ν ν LHS. Correspondingly, apart from the scalar field mediated force the neutrinos feel the gravitational drag exerted by the potential wells formed by CDM. Consequently, as long as thecoupling functionβ doesnotconsiderably enhance theinfluence of theterm GeffΩ δ , ∝ G ν ν thenon-relativistic neutrinos willfollowCDM(like baryons) just asinthe StandardModel. In the following we classify the behaviour of the neutrino density contrast in models of neutrino dark energy subject to all relevant kinds of coupling functions β. We emphasize that this classification is completely model independent. In the small-scale limit we distinguish the following three cases: Small-scale limit a) For Ω (1 + 2β2M2) < Ω until the present time, the neutrino density contrast ν pl CDM is stabilised by the CDM source term which dominates its dynamics. In this case the influence of the scalar field on the perturbations is subdominant and the density contrast in MaVaNs grows moderately just like gravitational instabilities in uncoupled neutrinos. b) For β const. and much larger than all other parameters at late times, G G, eff ∼ ≫ ˙ the damping term Hδ in Eq. (26) as well as the the terms proportional to δ and ν CDM δ can be neglected, leading to exponentially growing solutions. b c) For β = const. and growing faster than all other parameters at late times, G G, eff 6 ≫ δ is growing faster than exponentially . ν ¶ Note that if the upper limit from the Mainz experiment is saturated the requirement Ω Ω is ν m k ≪ formally not satisfied. However, this case should be viewed as very extreme and is most likely excluded based on structure formation arguments In the limit β(τ) for τ , Eq. 26 takes the form δ¨ 3H2Ω β2(τ)Mp2l δ = 0, and it ¶ → ∞ → ∞ ν − ν1+a2(Vφ′′+ρνβ′)/k2 ν canbe shownthat δ˙ν for τ [79]. Since this ratiois constantandthus notlargeenoughfor an |δν|→∞ →∞ exponentially growing δ , the solution is required to grow faster than exponentially. ν Neutrino Dark Energy – Revisiting the Stability Issue 10 In contrast, on scales (V′′ +ρ β′)−1/2 a/k < H−1 much larger than the range of the φ ν ≪ φ-mediated force, Large-scale limit d) For β const. and of moderate strength, G G and the perturbations behave eff ∼ ∼ effectively like perturbations for uncoupled fluids in General Relativity. e) For β growing faster than all other quantities at late times, G G, instabilities eff ≫ develop on all sub-Hubble scales a/k > (V′′ +ρ β′)−1/2 according to c). However, on φ ν large length scales their growth rate is suppressed due to the corresponding small wave number k. 3.1. Potentials and Couplings In the following, we consider two combinations of scalar potentials V (φ) and of scalar- φ neutrino couplings β which define our MaVaN models. The potentials are chosen to accomplish the required cosmic late-time acceleration and for the couplings we take meaningful limiting cases. Our main point is to present a proof of concept of the stability conditions stated above, which is valid for a general adiabatic MaVaN model. We note that a certain degree of fine-tuning is exerted. It is mainly due to the fact that CMBFAST and CAMB only operate in the linear regime H 10−4Mpc−1 < k < 0.1Mpc−1 and correspondingly, only ∼ in this regime can we analytically track the evolution of perturbation by the help of linear theory. Since the Compton wavelength of the scalar field m−1 sets the length scale of ∼ φ interest where possible instabilities can grow fastest, a/k > m−1, (cf. the discussion in φ sec. 2), this implies the scalar field mass has to be tuned a∼ccordingly, while at the same time the correct cosmology has to be accomplished. Firstly, we consider a MaVaN model suggested by [38] which we will refer to as the log-linear model. The scalar field has a Coleman-Weinberg type [78] logarithmic potential, V (φ) = V log(1+κφ), (29) φ 0 where the constants V and κ are chosen appropriately to yield Ω 0.7 and m H 0 DE φ ∼ ≫ today. The choice of V determines the evolution of φ according to Eq. (7) as plotted in φ fig.1. Apparently, theneutrino background hasastabilising effect onφ. It drives thescalar fieldtolargervaluesandstopsitfromrollingdownitspotentialV . Thiscompetitionofthe φ two terms in Eq. (7) results in a minimum at an intermediate value of φ (cf. Eq. 9), which slowly evolves due to changes in the neutrino energy density. As the universe expands and ρ dilutes, both the minimum and the scalar field are driven to smaller values towards zero. ν Let us now turn to the neutrino mass and its evolution. The dependence of m on the ν scalar field is given by, φ 0 m (φ) = m . (30) ν 0 φ